A Provable Oracle-Free Quantum Algorithm for Nonlinear Dynamics on Hybrid Oscillator-Qubit Processors

We develop a hybrid qubit--qumode algorithm for nonlinear ordinary differential equations of the form $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ with drift of polynomial degree~$L$. Following the Fokker--Planck route of Tennie and Magri, the algorithm propagates the state density and returns the deterministic trajectory as the peak of that density in the small-noise limit. The discretised generator is carried into a parametrised family of Schr\"{o}dinger equations by the warped-phase transformation of Jin, Liu, and Yu, and the Fourier-mode parameter of that family is placed on a single continuous-variable qumode. Our central structural result is that the Hermitian parts $H_{1}$ and $H_{2}$ of the discretised generator admit a bipartite Pauli decomposition that sorts the non-zero Pauli strings into $\mathcal{O}(\log N)$ mutually commuting families and factorises each family into a diagonal of degree at most $L$ tensored with a fixed rank-two bond operator. The factorisation renders each family exponential an exact product of $\mathcal{O}(n^{L})$ monomial-controlled momentum displacements, with no intra-family Trotter error. On a $d$-dimensional grid of $N=2^{n}$ points per axis the circuit costs $\mathcal{O}(d^{L+1}n^{L+2})$ gates per Trotter step. No sparse-access oracle and no block encoding is invoked: every gate is fixed in closed form by the polynomial coefficients of the drift. We also prove a bound on the numerical abscissa $\lambda_{\max}(H_{1})$ that fixes the recovery domain of the warped-phase transform and the post-selection cost. A classical simulation on two nonlinear benchmarks confirms the structural theorems, the shifted recovery, and the accuracy-per-resource advantage of the continuous-variable coupling over a discretised mode register.